If condition REF  does not hold for the linearised \(H\) -action on \(Y\) , then as explained in [1]} one can instead perform a sequence of equivariant blow-ups in order to arrange that condition REF  holds on the blow-up. These blow-ups are similar to the blow up sequences used to construct partial desingularisations of reductive GIT quotients (cf. \(§\)REF  and [2]}), but where one now considers unipotent stabiliser groups instead. Away from the exceptional locus, we can identify the semistable locus in the blown up space \(\widetilde{Y}\)  with a certain open subset of \(Y\) ; hence one obtains a quasi-projective quotient of this open subset, and a compactification given by the quotient of \(\widetilde{Y}\) . To identify this open subset in \(Y\) , one has to describe the complement to the exceptional divisor in the \(\widehat{U}\) -stable locus for \(\widetilde{Y}\) . This is more difficult when all of \(Z_{\min }\)  is blown up (see [3]} for further discussion).
